Question

Air at \(15^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) flows over a \(0.3-\mathrm{m}\)-wide plate at \(65^{\circ} \mathrm{C}\) at a velocity of $3.0 \mathrm{~m} / \mathrm{s}$. Compute the following quantities at \(x=x_{\mathrm{cr}}\) : (a) Hydrodynamic boundary layer thickness, \(\mathrm{m}\) (b) Local friction coefficient (c) Average friction coefficient (d) Total drag force due to friction, \(\mathrm{N}\) (e) Local convection heat transfer coefficient, W/m² \(\mathrm{K}\) (f) Average convection heat transfer coefficient, W/m². \(\mathrm{K}\) (g) Rate of convective heat transfer, W

Short answer

Answer: In order to find the rate of convective heat transfer for a laminar flow over a flat plate, the following parameters need to be calculated: a) Hydrodynamic boundary layer thickness at the critical location: \(\delta(x_{cr}) = 5.0 \times \frac{x_{cr}}{\sqrt{Re_{x_{cr}}}}\) b) Local friction coefficient: \(C_f = \frac{0.664}{\sqrt{Re_{x_{cr}}}}\) c) Average friction coefficient: \(C_{f,avg} = \frac{1.328}{\sqrt{Re_{x_{cr}}}}\) d) Total drag force due to friction: \(F_{D} = \frac{1}{2}\rho U_{\infty}^{2}C_{f,avg}\cdot A\) e) Local convection heat transfer coefficient: \(h_{x} = 0.332 \cdot k \cdot \frac{U_{\infty}}{x_{cr}} \cdot Re_{x_{cr}}^{1/2} \cdot Pr^{1/3}\) f) Average convection heat transfer coefficient: \(h_{avg} = 0.664 \cdot k \cdot \frac{U_{\infty}}{L} \cdot Re_{L}^{1/2} \cdot Pr^{1/3}\) g) Rate of convective heat transfer: \(q = h_{avg} \cdot A \cdot (T_s - T_{\infty})\)

Step-by-step solution

Determine the properties of air at the mean film temperature.

First, we need to find the mean film temperature, \(T_{film}\), which is the average of the air temperature \(T_{\infty}\) and plate temperature \(T_s\). Then, we can use the mean film temperature to obtain the properties of air: \(T_{\infty} = 15^{\circ} \mathrm{C}\), \(T_s = 65^{\circ} \mathrm{C}\), \(U_{\infty} = 3.0\) m/s, and \(T_{film} = \frac{T_{\infty} + T_s}{2}\).

Calculate the Reynolds number at the critical location.

The critical Reynolds number, \(Re_{x_{cr}}\), is used to identify the onset of turbulence in the boundary layer flow. It can be calculated using the following formula: \(Re_{x_{cr}} = \frac{U_{\infty} x_{cr}}{\nu}\), where \(x_{cr}\) is the critical location, and \(\nu\) is the kinematic viscosity of the air at the film temperature. If the Reynolds number is less than the critical value, the flow is considered to be laminar.

Calculate the hydrodynamic boundary layer thickness at the critical location.\((a)\)

Once the Reynolds number at the critical location is determined, we can calculate the hydrodynamic boundary layer thickness, \(\delta(x_{cr})\), using the formula for laminar flow: \(\delta(x_{cr}) = 5.0 \times \frac{x_{cr}}{\sqrt{Re_{x_{cr}}}}\).

Calculate the local friction coefficient \((b)\)

The local friction coefficient, \(C_f\), can be calculated using the following formula for the laminar flow: \(C_f = \frac{0.664}{\sqrt{Re_{x_{cr}}}}\).

Calculate the average friction coefficient \((c)\)

The average friction coefficient, \(C_{f,avg}\), can be calculated using the following formula for the laminar flow: \(C_{f,avg} = \frac{1.328}{\sqrt{Re_{x_{cr}}}}\).

Calculate the total drag force due to friction \((d)\)

To calculate the total drag force due to friction, \(F_{D}\), we can use the following formula: \(F_{D} = \frac{1}{2}\rho U_{\infty}^{2}C_{f,avg}\cdot A\), where \(A\) is the total area of the plate in contact with the air flow.

Calculate the local convection heat transfer coefficient \((e)\)

The local convection heat transfer coefficient, \(h_{x}\), can be calculated using the following formula for laminar flow: \(h_{x} = 0.332 \cdot k \cdot \frac{U_{\infty}}{x_{cr}} \cdot Re_{x_{cr}}^{1/2} \cdot Pr^{1/3}\), where \(k\) is the thermal conductivity of the air, and \(Pr\) is the Prandtl number for the air.

Calculate the average convection heat transfer coefficient \((f)\)

To calculate the average convection heat transfer coefficient, \(h_{avg}\), we can use the following formula for laminar flow: \(h_{avg} = 0.664 \cdot k \cdot \frac{U_{\infty}}{L} \cdot Re_{L}^{1/2} \cdot Pr^{1/3}\), where \(L\) is the total length of the plate.

Calculate the rate of convective heat transfer \((g)\)

Finally, we can calculate the rate of convective heat transfer, \(q\), using the following formula: \(q = h_{avg} \cdot A \cdot (T_s - T_{\infty})\).